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d5b7741 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 | import OAI.Combinatorics.ContingencyTables.Transport.GraphCompletionCounts
namespace OAI
/-!
# The companion's actual small/free profiles and tree intervals
Residual margins subtract the two separate small views and the fixed
free values. The integral graph completion satisfies every nonroot line.
The maximum-forest cut bound then supplies, rather than assumes, the
physical tree interval used by the signature/count marginal.
-/
namespace ContingencyTables.ForestPhysicalCompletion
open scoped BigOperators Classical
open SimpleGraph
open ForestCapacity GraphIntegralCompletion GraphCompletionCounts
variable {I J : Type*} [Fintype I] [Fintype J]
noncomputable def fixedRow (small : I → J → Bool) (T : SimpleGraph (I⊕J))
(x free : I → J → ℤ) (i : I) (j : J) : ℤ :=
if small i j then x i j else if T.Adj (.inl i) (.inr j) then 0 else free i j
noncomputable def fixedColumn (small : I → J → Bool) (T : SimpleGraph (I⊕J))
(q free : I → J → ℤ) (i : I) (j : J) : ℤ :=
if small i j then q i j else if T.Adj (.inl i) (.inr j) then 0 else free i j
noncomputable def rowResidual (small : I → J → Bool) (T : SimpleGraph (I⊕J))
(r : I → ℤ) (x free : I → J → ℤ) (i : I) : ℤ :=
r i-∑ j, fixedRow small T x free i j
noncomputable def columnResidual (small : I → J → Bool) (T : SimpleGraph (I⊕J))
(c : J → ℤ) (q free : I → J → ℤ) (j : J) : ℤ :=
c j-∑ i, fixedColumn small T q free i j
noncomputable def solved (small : I → J → Bool) (T : SimpleGraph (I⊕J))
(r : I → ℤ) (c : J → ℤ) (x q free : I → J → ℤ) : I → J → ℤ :=
tableCompletion T (rowResidual small T r x free) (columnResidual small T c q free)
noncomputable def completed (small : I → J → Bool) (T : SimpleGraph (I⊕J))
(r : I → ℤ) (c : J → ℤ) (x q free : I → J → ℤ) (i : I) (j : J) : ℤ :=
if T.Adj (.inl i) (.inr j) then solved small T r c x q free i j else free i j
lemma row_pointwise (small : I → J → Bool) (T : SimpleGraph (I⊕J))
(hTG : T≤cellGraph small) (r : I → ℤ) (c : J → ℤ) (x q free : I → J → ℤ)
(i : I) (j : J) :
(if small i j then x i j else completed small T r c x q free i j)=
solved small T r c x q free i j+fixedRow small T x free i j := by
by_cases ht : T.Adj (.inl i) (.inr j)
· have hs : small i j=false := hTG ht
simp only [hs,Bool.false_eq_true,ite_false,completed,ht,ite_true,fixedRow,add_zero]
· have hz : solved small T r c x q free i j=0 := tableCompletion_supported T _ _ i j ht
simp only [completed,ht,ite_false,hz,zero_add,fixedRow]
lemma column_pointwise (small : I → J → Bool) (T : SimpleGraph (I⊕J))
(hTG : T≤cellGraph small) (r : I → ℤ) (c : J → ℤ) (x q free : I → J → ℤ)
(i : I) (j : J) :
(if small i j then q i j else completed small T r c x q free i j)=
solved small T r c x q free i j+fixedColumn small T q free i j := by
by_cases ht : T.Adj (.inl i) (.inr j)
· have hs : small i j=false := hTG ht
simp only [hs,Bool.false_eq_true,ite_false,completed,ht,ite_true,fixedColumn,add_zero]
· have hz : solved small T r c x q free i j=0 := tableCompletion_supported T _ _ i j ht
simp only [completed,ht,ite_false,hz,zero_add,fixedColumn]
lemma row_completed_sum (small : I → J → Bool) (T : SimpleGraph (I⊕J))
(hTG : T≤cellGraph small) (r : I → ℤ) (c : J → ℤ) (x q free : I → J → ℤ)
(i : I) (hi : Sum.inl i≠componentRoot T (.inl i)) :
(∑ j, if small i j then x i j else completed small T r c x q free i j)=r i := by
simp_rw [row_pointwise small T hTG r c x q free]
rw [Finset.sum_add_distrib,show (∑ j, solved small T r c x q free i j)=
rowResidual small T r x free i from tableCompletion_row small T hTG _ _ i hi]
exact sub_add_cancel _ _
lemma column_completed_sum (small : I → J → Bool) (T : SimpleGraph (I⊕J))
(hTG : T≤cellGraph small) (r : I → ℤ) (c : J → ℤ) (x q free : I → J → ℤ)
(j : J) (hj : Sum.inr j≠componentRoot T (.inr j)) :
(∑ i, if small i j then q i j else completed small T r c x q free i j)=c j := by
simp_rw [column_pointwise small T hTG r c x q free]
rw [Finset.sum_add_distrib,show (∑ i, solved small T r c x q free i j)=
columnResidual small T c q free j from tableCompletion_column small T hTG _ _ j hj]
exact sub_add_cancel _ _
omit [Fintype I] [Fintype J] in
lemma nonroot_of_root_component (T : SimpleGraph (I⊕J)) (u v : I⊕J)
(h : T.Reachable (componentRoot T u) v) (hv : v≠componentRoot T u) :
v≠componentRoot T v := by
have he : componentRoot T v=componentRoot T u := by
rw [← componentRoot_constant T h,componentRoot_idempotent]
simpa only [he] using hv
/-- The completion interval holds for the literal residual
margin equations, including every box-valid unbalanced small profile. -/
theorem solved_in_intervals (small : I → J → Bool) (b : I → J → ℕ)
(T : SimpleGraph (I⊕J)) (hTG : T≤cellGraph small) (hT : T.IsAcyclic)
(hmax : ∀ F : SimpleGraph (I⊕J), F≤cellGraph small → F.IsAcyclic →
MaximumForest.weight F (capacity b)≤MaximumForest.weight T (capacity b))
(r : I → ℤ) (c : J → ℤ) (x q free X : I → J → ℤ) (d W : ℕ)
(hd : 1≤d) (hcard : Fintype.card I*Fintype.card J≤d)
(hlarge : ∀ i j, small i j=false → W≤b i j ∧ 1≤b i j)
(hsmall : ∀ i j, small i j=true → b i j<W ∧
(0≤x i j ∧ x i j≤b i j) ∧ (0≤q i j ∧ q i j≤b i j))
(hfree : ∀ i j, small i j=false → ¬T.Adj (.inl i) (.inr j) →
0≤free i j ∧ free i j≤b i j)
(hX : ∀ i j, 0≤X i j ∧ X i j≤b i j)
(hXr : ∀ i, (∑ j, X i j)=r i) (hXc : ∀ j, (∑ i, X i j)=c j) :
∀ i j, T.Adj (.inl i) (.inr j) →
-((10*d*b i j:ℕ):ℤ) ≤ solved small T r c x q free i j ∧
solved small T r c x q free i j≤(10*d*b i j:ℕ) := by
intro i₀ j₀ ht
let root := componentRoot T (.inl i₀)
have hb := completion_interval small b T hTG hT hmax i₀ j₀ ht root
(componentRoot_reachable T (.inl i₀)).symm
(fun i j => (x i j:ℝ)) (fun i j => (q i j:ℝ))
(fun i j => (completed small T r c x q free i j:ℝ)) (fun i j => (X i j:ℝ))
(d:ℝ) W (by exact_mod_cast hd) (by exact_mod_cast hcard) hlarge
(by
intro i j hs
exact ⟨(hsmall i j hs).1,by exact_mod_cast (hsmall i j hs).2.1,
by exact_mod_cast (hsmall i j hs).2.2⟩)
(by intro i j; exact_mod_cast hX i j)
(by
intro i j hs htree
have hn : ¬T.Adj (.inl i) (.inr j) := by
simpa only [treeCell,decide_eq_false_iff_not] using htree
simp only [completed,hn,ite_false]
have hf := hfree i j hs hn
constructor
· exact_mod_cast hf.1
· have hcast : (free i j:ℝ)≤(b i j:ℝ) := by exact_mod_cast hf.2
linarith)
(by
intro i hcomp hi
have hrow := row_completed_sum small T hTG r c x q free i
(nonroot_of_root_component T (.inl i₀) (.inl i) hcomp hi)
have hrowR : (∑ j, if small i j then (x i j:ℝ) else
(completed small T r c x q free i j:ℝ))=(r i:ℝ) := by exact_mod_cast hrow
have hXR : (∑ j, (X i j:ℝ))=(r i:ℝ) := by exact_mod_cast hXr i
unfold ForestCut.rowDifference
calc
_=(∑ j, if small i j then (x i j:ℝ) else
(completed small T r c x q free i j:ℝ))-(∑ j, (X i j:ℝ)) := by
rw [← Finset.sum_sub_distrib]
apply Finset.sum_congr rfl
intro j _
split_ifs <;> rfl
_=0 := by rw [hrowR,hXR,sub_self])
(by
intro j hcomp hj
have hcol := column_completed_sum small T hTG r c x q free j
(nonroot_of_root_component T (.inl i₀) (.inr j) hcomp hj)
have hcolR : (∑ i, if small i j then (q i j:ℝ) else
(completed small T r c x q free i j:ℝ))=(c j:ℝ) := by exact_mod_cast hcol
have hXC : (∑ i, (X i j:ℝ))=(c j:ℝ) := by exact_mod_cast hXc j
unfold ForestCut.columnDifference
calc
_=(∑ i, if small i j then (q i j:ℝ) else
(completed small T r c x q free i j:ℝ))-(∑ i, (X i j:ℝ)) := by
rw [← Finset.sum_sub_distrib]
apply Finset.sum_congr rfl
intro i _
split_ifs <;> rfl
_=0 := by rw [hcolR,hXC,sub_self])
simp only [completed,ht,ite_true] at hb
constructor
· have hlow : (-10:ℤ)*(d:ℤ)*(b i₀ j₀:ℤ) ≤ solved small T r c x q free i₀ j₀ := by
exact_mod_cast hb.1
convert hlow using 1
push_cast
ring
· exact_mod_cast hb.2
/-- For the literal small/free residual margins, the equality-limit
sum over the prescribed tree intervals retains exactly one term, with
precisely the weight of the source's solved tree values. -/
theorem tree_indicator_sum (small : I → J → Bool) (b : I → J → ℕ)
(T : SimpleGraph (I⊕J)) (hTG : T≤cellGraph small) (hT : T.IsAcyclic)
(hmax : ∀ F : SimpleGraph (I⊕J), F≤cellGraph small → F.IsAcyclic →
MaximumForest.weight F (capacity b)≤MaximumForest.weight T (capacity b))
(r : I → ℤ) (c : J → ℤ) (x q free X : I → J → ℤ) (d W : ℕ)
(hd : 1≤d) (hcard : Fintype.card I*Fintype.card J≤d)
(hlarge : ∀ i j, small i j=false → W≤b i j ∧ 1≤b i j)
(hsmall : ∀ i j, small i j=true → b i j<W ∧
(0≤x i j ∧ x i j≤b i j) ∧ (0≤q i j ∧ q i j≤b i j))
(hfree : ∀ i j, small i j=false → ¬T.Adj (.inl i) (.inr j) →
0≤free i j ∧ free i j≤b i j)
(hX : ∀ i j, 0≤X i j ∧ X i j≤b i j)
(hXr : ∀ i, (∑ j, X i j)=r i) (hXc : ∀ j, (∑ i, X i j)=c j)
(weight : (I → J → ℤ) → ℝ) :
(∑ a : TreeIndex T d b,
if NonrootMargins T (rowResidual small T r x free) (columnResidual small T c q free)
(physical T d b a) then weight (physical T d b a) else 0)=
weight (solved small T r c x q free) := by
exact weighted_completion_sum small T hTG hT (rowResidual small T r x free)
(columnResidual small T c q free) d b
(solved_in_intervals small b T hTG hT hmax r c x q free X d W hd hcard hlarge
hsmall hfree hX hXr hXc) weight
end ContingencyTables.ForestPhysicalCompletion
end OAI
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